Remark 4.1 We notice that the algorithms with or without \(q\)  should have very similar behaviours, because \(\mathcal {A}_{I,q}\)  do exactly the same predictions on \(I\)  whatever \(q\)  is. Introducing the index \(q\)  is for the ease of analysis and only loses a small \(\log (T)\)  factor compared with the optimal full-matrix bound [1]}.

